Factorization à la Dirac Applied to Some Equations of Classical Physics

Zine El Abiddine Fellah, Erick Ogam, Mohamed Fellah and Claude Depollier
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Zine El Abiddine Fellah: LMA, CNRS, UPR 7051, Centrale Marseille, Aix-Marseille University, CEDEX 20, F-13402 Marseille, France
Erick Ogam: LMA, CNRS, UPR 7051, Centrale Marseille, Aix-Marseille University, CEDEX 20, F-13402 Marseille, France
Mohamed Fellah: Laboratoire de Physique Théorique, Faculté de Physique, USTHB, BP 32 El Alia, Bab Ezzouar 16111, Algeria
Claude Depollier: UMR CNRS 6613, Laboratoire d’Acoustique de l’Universite du Maine, LUNAM Universite du Maine, UFR STS Avenue O. Messiaen, CEDEX 09, 72085 Le Mans, France

Mathematics, 2021, vol. 9, issue 8, 1-14

Abstract: In this paper, we present an application of Dirac’s factorization method to three types of the partial differential equations, i.e., the wave equation, the scattering equation, and the telegrapher’s equation. This method gives results that contribute to a better understanding of physical phenomena by generalizing the Euler and constituent equations. Its application to the wave equation shows that it is indeed a factorization method, since it gives d’Alembert’s solutions in a more general framework. In the case of the diffusion equation, a fractional differential equation has been established that has already been highlighted by other authors in particular cases, but by indirect methods. Dirac’s method brings several new results in the case of the telegraphers’ equation corresponding to the propagation of an acoustic wave in a dissipative fluid. On the one hand, its formalism facilitates the temporal interpretation of phenomena, in particular the density and compressibility of the fluid become temporal operators, which can be “seen” as susceptibilities of the fluid. On the other hand, a consequence of this temporal modeling is the highlighting in Euler’s equation of a term similar to the one that was introduced by Boussinesq and Basset in the equation of the motion of a solid sphere in a unsteady fluid.

Keywords: factorization method; diffusion equation; wave equation; telegraph equation; fractional calculus; basset force (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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